Latent Variables
Overview
The latent variable approach is a mathematical trick which allows one to compute credit default statistics easily,
especially when correlations among defaults are involved. While strictly speaking, the latent variables introduced into
the problem need not have an economic interpretation, they can be set within the framework of the
merton model,
where the latent variable is interpreted as the asset value of the firm, or a systemic risk variable that drives
the asset values of the firms being considered.
Latent Variables
In order to simplify the calculations, modelers often model defaults using laten variables. That is, we assume there
are other random variables for which the default variables are tied. For example, we can assume that there is
a set of random variables {% A_i %} such that the following holds.
{% y_i = 1 %} when {% A_i \leq d_i %}
and
{% prob_i = \Phi(d_i) %}
or
{% d_i = \Phi^{-1}(prob_i) %}
{% y_i = 0 %} when {% A_i > d_i %}
Then the joint probability of default can be modeled as
{% Prob(y_1 = 1, y_j = 1) = Prob(A_i \leq d_i \: and \: A_j \leq d_j) %}
Typically, the {% A_i %} variables are modeled as
{% A_i = w_i Z + \sqrt{1-w_i^2} e_i %}
where both {% Z %} and {% e_i %} are
normally distributed
random variables with mean 0 and variance equal to 1, and with
{% cov(e_i, e_j) = 0 %}
and also
{% cov(e_i, Z) = 0 %}
Calculating Probability of Joint Default
The correlation of the latent variables is given by
{% \rho_{ij}^{asset} = \frac{cov(A_i, A_j)}{\sigma_{Ai} \sigma_{Aj}} %}
{% \rho_{ij}^{asset} = \frac{cov(w_i Z + \sqrt{1-w_i^2} e_i , w_j Z + \sqrt{1-w_j^2} e_j)}{1 \times 1} %}
{% \rho_{ij}^{asset} = w_i \times w_j %}
{% Prob(A_i \leq d_i , A_j \leq d_j) = prob_{ij} = \Phi_2(d_i,d_j, \rho_{ij}^{asset}) %}
where {% \Phi %} is the standard bivariate normal distribution function with the given correlation.
If {% prob_{ij} %}, {% prob_i %} and {% prob_j %} are known, then {% \rho_{ij}^{asset} %} can be found using such
methods as
Root Finding. That is we have
{% \Phi_2(d_i,d_j, \rho_{ij}^{asset}) - prob_{ij} = 0 %}
and we seek to find {% \rho_{ij}^{asset} %} that makes this equation true.
Topics
